Optimal. Leaf size=44 \[ \frac{1}{3} \sqrt{-x^4+x^2+2} x+F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.130613, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{1}{3} \sqrt{-x^4+x^2+2} x+F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2 + x^2 - x^4],x]
[Out]
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Rubi in Sympy [A] time = 19.9534, size = 42, normalized size = 0.95 \[ \frac{x \sqrt{- x^{4} + x^{2} + 2}}{3} + \frac{E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{3} + F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**4+x**2+2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0855372, size = 90, normalized size = 2.05 \[ \frac{-x^5+x^3-3 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+2 x}{3 \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[2 + x^2 - x^4],x]
[Out]
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Maple [B] time = 0.004, size = 125, normalized size = 2.8 \[{\frac{x}{3}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{2\,\sqrt{2}}{3}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{\sqrt{2}}{6}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^4+x^2+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} + x^{2} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^4 + x^2 + 2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{-x^{4} + x^{2} + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^4 + x^2 + 2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- x^{4} + x^{2} + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**4+x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} + x^{2} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^4 + x^2 + 2),x, algorithm="giac")
[Out]